Dynamic equations on time scales: a surveyweb.mst.edu/~bohner/papers/deotsas.pdfThe study of dynamic...

Journal of Computational and Applied Mathematics 141 (2002) 1–26www.elsevier.com/locate/cam

Dynamic equations on time scales: a surveyRavi Agarwala, Martin Bohnerb; ∗, Donal O’Reganc, Allan Petersond

aNational University of Singapore, Department of Mathematics, Singapore 119260bUniversity of Missouri-Rolla, Department of Mathematics and Statistics, Rolla, MO 65409-0020, USA

cNational University of Ireland, Department of Mathematics, Galway, IrelanddUniversity of Nebraska-Lincoln, Department of Mathematics and Statistics, Lincoln, NE 68588-0323, USA

Received 20 August 2000; received in revised form 26 February 2001

Abstract

The study of dynamic equations on time scales, which goes back to its founder Stefan Hilger (1988), is an area ofmathematics that has recently received a lot of attention. It has been created in order to unify the study of di6erential anddi6erence equations. In this paper we give an introduction to the time scales calculus. We also present various propertiesof the exponential function on an arbitrary time scale, and use it to solve linear dynamic equations of 9rst order. Severalexamples and applications, among them an insect population model, are considered. We then use the exponential functionto de9ne hyperbolic and trigonometric functions and use those to solve linear dynamic equations of second order withconstant coe:cients. Finally, we consider self-adjoint equations and, more generally, so-called symplectic systems, andpresent several results on the positivity of quadratic functionals. c© 2002 Elsevier Science B.V. All rights reserved.

MSC: 34A40; 39A13

Keywords: Time scales; Dynamic equations; Measure chains

1. Unifying continuous and discrete analysis

In 1988, Stefan Hilger [12] introduced the calculus of measure chains in order to unify continuousand discrete analysis. Bernd Aulbach, who supervised Stefan Hilger’s Ph.D. thesis [11], points outthe three main purposes of this new calculus:

Uni9cation – Extension – Discretization:

For many purposes in analysis it is su:cient to consider a special case of a measure chain, aso-called time scale, which simply is a closed subset of the real numbers. We denote a time scale

∗ Corresponding author.E-mail addresses: [email protected] (R. Agarwal), [email protected] (M. Bohner), [email protected]

(D. O’Regan), [email protected] (A. Peterson).

0377-0427/02/$ - see front matter c© 2002 Elsevier Science B.V. All rights reserved.PII: S 0377-0427(01)00432-0

2 R. Agarwal et al. / Journal of Computational and Applied Mathematics 141 (2002) 1–26

by the symbol T. For functions y de9ned on T; we can consider a so-called delta derivative yG;and this delta derivative is equal to y′ (the usual derivative) if T=R is the set of all real numbers,and it is equal to Gy (the usual forward di6erence) if T=Z is the set of all integers. Then we canstudy dynamic equations

f(t; y; yG; yG2; : : : ; yGn

) = 0;

which may also involve higher order derivatives as indicated. Along with such dynamic equationswe can consider initial value problems and boundary value problems. We remark that these dynamicequations are di6erential equations when T=R; and they are di6erence equations when T=Z. Otherkinds of equations are covered by them as well, such as for example q-di8erence equations

T= qZ:={qk | k ∈Z} ∪ {0} for some q¿ 1

or di6erence equations with constant step size

T= hZ:={hk | k ∈Z} for some h¿ 0:

Particularly useful for the discretization aspect are time scales of the form

T= {tk | k ∈Z} where tk ∈R; tk ¡ tk+1 for all k ∈Z:This survey paper is organized as follows: In Section 2 we introduce the basic concepts of thetime scales calculus. One major object, when studying dynamic equations on time scales, is theexponential function, which will be discussed in Section 3. Section 4 is devoted to examples andapplications. In Section 5 we consider linear dynamic equations and initial value problems involvingthem. Finally, in Section 6, we study a particular case of linear dynamic systems, namely symplecticsystems on time scales. The bibliography at the end of this paper contains, besides references thatare cited in the text, many of the recent articles in this relatively new area of research. The book[7] is an up-to-date summary on the subject.

2. Delta derivatives

A time scale is a nonempty closed subset of the reals, and we usually denote it by the symbol T.The two most popular examples are T=R and T=Z. We de9ne the forward and backward jumpoperators �; : T→ T by

�(t) = inf{s∈T | s¿ t} and (t) = sup{s∈T | s¡ t}(supplemented by inf ∅= supT and sup ∅= inf T). A point t ∈T is called right-scattered, right-dense,left-scattered, left-dense, if �(t)¿t; �(t) = t; (t)¡t; (t) = t holds, respectively. The set T� is de-9ned to be T if T does not have a left-scattered maximum; otherwise it is T without this left-scatteredmaximum. The graininess � : T→ [0;∞) is de9ned by

�(t) = �(t) − t:

Hence the graininess function is constant 0 if T=R while it is constant 1 for T=Z. However, atime scale T could have nonconstant graininess. Now, let f be a function de9ned on T. We say

R. Agarwal et al. / Journal of Computational and Applied Mathematics 141 (2002) 1–26 3

that f is delta di8erentiable (or simply: di8erentiable) at t ∈T� provided there exists an � suchthat for all �¿ 0 there is a neighborhood N around t with

|f(�(t)) − f(s) − �(�(t) − s)|6 �|�(t) − s| for all s∈N:

In this case we denote the � by fG(t); and if f is di6erentiable for every t ∈T�; then f is said tobe di8erentiable on T and fG is a new function de9ned on T�. If f is di6erentiable at t ∈T�; thenit is easy to see that

fG(t) =

lims→t; s∈Tf(t) − f(s)

t − sif �(t) = 0

f(�(t)) − f(t)�(t)

if �(t)¿ 0:(2.1)

However, it is exactly the philosophy of the calculus on time scales to avoid the separate discussionof the two cases �(t) = 0 and �(t)¿ 0, i.e., when t is right-dense and right-scattered, respectively.Results as in (2.1) do not serve this purpose and therefore should be avoided in proofs. To illustratethe idea, we now give another formula, which holds whenever f is di6erentiable at t ∈T�:

f(�(t)) =f(t) + �(t)fG(t): (2.2)

When applying formula (2.2), we do not need to distinguish between the two cases �(t) = 0 and�(t)¿ 0. Formula (2.2) holds in both of these cases. Even though it is trivial in the case of aright-dense t; we need not worry about di6erent cases. Two further examples of such formulas arethe product rule for the derivative of the product of two di6erentiable functions f and g:

(fg)G(t) =fG(t)g(t) + f(�(t))gG(t) (2.3)

and the quotient rule for the derivative of the quotient of two di6erentiable functions f and g = 0:(fg

)G

(t) =fG(t)g(t) − f(t)gG(t)

g(t)g(�(t)): (2.4)

Clearly, 1G = 0 and tG = 1; so we can use (2.3) to 9nd

(t2)G = (t · t)G = t + �(t);

and we can use (2.4) to 9nd(1t

)G

= − 1t�(t)

:

Other formulas may be obtained likewise. One word of caution: The forward jump operator � is notnecessarily a di6erentiable function. Clearly, at a point which is left-dense and right-scattered at thesame time, � is not continuous. Hence � is not di6erentiable at such a point since we know:


Theorem 2.1. Every di8erentiable function is continuous.

However, � is an example of a function which we call rd-continuous. A function f de9ned onT is rd-continuous, if it is continuous at every right-dense point and if the left-sided limit existsin every left-dense point. The importance of rd-continuous functions is revealed by the followingexistence result by Hilger [12]:

Theorem 2.2. Every rd-continuous function possesses an antiderivative.

Here, F is called an antiderivative of a function f de9ned on T if FG =f holds on T�. In thiscase we de9ne an integral by∫ t

sf(�)G�=F(t) − F(s) where s; t ∈T:

An antiderivative of 0 is 1, an antiderivative of 1 is t; but it is not possible to 9nd a polynomial(or any “nice” formula of a function) which is an antiderivative of t (where T is an arbitrary timescale). The role of t2 is therefore played in the time scales calculus by∫ t

0�(�)G� and

∫ t

0�G�:

Note that both integrals exist by Theorem 2.2 as the functions � and identity are both rd-continuous.In general, the functions

g0(t; s) ≡ 1 and gk+1(t; s) =∫ t

sgk(�(�); s)G�; k¿ 0

and

h0(t; s) ≡ 1 and hk+1(t; s) =∫ t

shk(�; s)G�; k¿ 0

may be considered as the “polynomials” on T. The relationship between gk and hk is given in [2]as

gk(t; s) = (−1)khk(t; s) for all k ∈N: (2.5)

To verify (2.5) e.g., for the case k = 2 we can calculate

g2(t; s) =∫ t

sg1(�(�); s)G�=

∫ t

s(�(�) − s)G�

=∫ t

s(�(�) + �)G�−

∫ t

s�G�−

∫ t

ssG�

=∫ t

s(�2)GG�+

∫ s

t�G�− s(t − s)

=∫ s

t�G�+ t2 − s2 − s(t − s)

=∫ s

t(�− t)G�=

∫ s

th1(�; t)G�= h2(s; t):


One of the two versions of Taylor’s formula (the other obtained by using (2.5)) reads as follows,see [2]:

Theorem 2.3. Let �∈T�n−1. If f is n times di8erentiable; then

f(t) =n−1∑k=0

hk(t; �)fGk(�) +

∫ n−1(t)

�hn−1(t; �(�))fGn

(�)G�:

3. Special functions

A function p : T→ R is called regressive if

1 + �(t)p(t) = 0 for all t ∈T:Concerning initial value problems

yG =p(t)y; y(t0) = 1 (3.1)

(where t0 ∈T; as is assumed throughout this paper), Hilger [12] proved the following existence anduniqueness theorem:

Theorem 3.1. If p is rd-continuous and regressive; then (3:1) has a unique solution.

We call the unique solution of (3.1) the exponential function and denote it by ep(·; t0). In fact,there is an explicit formula for ep(t; s); using the so-called cylinder transformation

�h(z) =

Log(1 + hz)z

if h = 0

z if h= 0:

The formula, see [13], reads

ep(t; s) = exp{∫ t

s��(�)(p(�))G�

}: (3.2)

It can be seen immediately from (3.2) that the exponential function never vanishes; however, incontrast to the case T=R, the exponential function could possibly attain negative values. As anexample, just consider the problem yG = − 2y; y(0) = 1 for T=Z. Now we will mention someproperties of the exponential function. First, we have by (2.2)

ep(�(t); s) = ep(t; s) + �(t)eGp (t; s)

= ep(t; s) + �(t)p(t)ep(t; s)

= [1 + �(t)p(t)]ep(t; s);


where we denote by eGp (t; s) the derivative of ep with respect to the 9rst variable. Using this equation

and letting y= ep(·; t0)eq(·; t0) for rd-continuous and regressive p and q, we 9nd by (2.3) that

yG(t) = eGp (t; t0)eq(t; t0) + ep(�(t); t0)eG

q (t; t0)

=p(t)ep(t; t0)eq(t; t0) + [1 + �(t)p(t)]ep(t; t0)q(t)eq(t; t0)

= [p(t) + q(t) + �(t)p(t)q(t)]y(t):

Hence, introducing an addition ⊕ by

p⊕ q=p + q + �pq;

we 9nd that y solves the initial value problem

yG = (p⊕ q)(t)y; y(t0) = 1;

and therefore y= ep⊕q(·; t0) by Theorem 3.1. Thus

ep⊕q = ep · eq:

Note also that p⊕ q is regressive i6 both p and q are:

1 + �(p⊕ q) = 1 + �(p + q + �pq) = 1 + �p + �q + �2pq= (1 + �p)(1 + �q):

To see how the corresponding subtraction � should be de9ned, we denote y= ep(·; t0)=eq(·; t0) anduse (2.4) to calculate

yG(t) =eGp (t; t0)eq(t; t0) − ep(t; t0)eG

q (t; t0)

eq(t; t0)eq(�(t); t0)

=p(t)ep(t; t0)eq(t; t0) − ep(t; t0)q(t)eq(t; t0)

eq(t; t0)[1 + �(t)q(t)]eq(t; t0)

=p(t) − q(t)1 + �(t)q(t)

y(t)

so that y solves the initial value problem

yG = (p� q)(t)y; y(t0) = 1

provided we introduce the subtraction � by

p� q=p− q1 + �q

:

Therefore, again by Theorem 3.1,

ep�q =epeq:


Note again that p� q is regressive if both p and q are:

1 + �(p� q) = 1 + �p− q1 + �q

=1 + �q + �(p− q)

1 + �q=

1 + �p1 + �q

:

We put

�q= 0 � q= − q1 + �q

;

and then we can derive many useful formulas such as e.g., �(�q) = q. We have

Theorem 3.2. The set of regressive functions together with addition ⊕ is an Abelian group.

Further properties of the exponential function may be obtained as in [8,13]. Next we want toemphasize, that, along with the equation yG =p(t)y, there is another “natural” form of a linearequation of 9rst order, namely xG =−p(t)x�, where we denote x� = x ◦�. Of course for T=R boththese equations are (up to a minus sign) the same. However, in the general time scales setting, thetwo equations are di6erent. Let us now solve the initial value problem

xG = − p(t)x�; x(t0) = 1; (3.3)

subject to the usual assumptions on p. Assuming that x solves (3.3), we 9nd by using (2.2)

xG(t) = − p(t)x(�(t)) = − p(t)[x(t) + �(t)xG(t)]

and hence

[1 + �(t)p(t)]xG(t) = − p(t)x(t)

so that (note that p is assumed to be regressive)

xG(t) = − p(t)1 + �(t)p(t)

x(t) = (�p)(t)x(t):

Therefore x is a solution of

yG = (�p)(t)y; y(t0) = 1:

Theorem 3.3. If p is rd-continuous and regressive; then ep(·; t0) and e�p(·; t0) are the unique solu-tions of (3:1) and (3:3); respectively.

Using the exponential function, we now may 9nd solutions of many dynamic equations in a similarway as we would do it with ordinary di6erential equations. As an example, we consider the equation

yG3 − 2yG2 − yG + 2y= 0:


Trying y(t) = e�(t; t0) with constant � leads us to

0 = (�3 − 2�2 − � + 2)e�(t; t0) = (� + 1)(�− 1)(�− 2)e�(t; t0);

i.e., �∈{−1; 1; 2} so that e−1(·; t0); e1(·; t0); and e2(·; t0) are solutions. All solutions of this dynamicequation may be constructed by taking linear combinations of the above three solutions. As anotherexample, we consider the initial value problem

yG2= a2y; y(t0) = 1; yG(t0) = 0; (3.4)

where a is a regressive constant. By using the technique described above, we 9nd that both ea(·; t0)and e−a(t; t0) solve the dynamic equation. Therefore we try a linear combination of the form�ea(·; t0) + �e−a(·; t0). We 9nd � + �= 1 and � − �= 0 so that �= �= 1=2. Hence it is usefulto introduce

coshp =ep + e−p

2and sinhp =

ep − e−p2

;

where sinh is the derivative of cosh. Both de9nitions require p and −p to be regressive (andrd-continuous), and because of

(1 + �p)(1 − �p) = 1 − �2p2;

this is equivalent to −�p2 being regressive. The following result is easy to verify:

Theorem 3.4. If p is rd-continuous and −�p2 is regressive; then

sinhGp = coshp; coshG

p = sinhp; and cosh2p − sinh2

p = e−�p2 :

Now we also introduce trigonometric functions in a similar fashion:

cosp =eip + e−ip

2and sinp =

eip − e−ip2i

;

provided �p2 is regressive, and this is clearly satis9ed if p(t)∈R for all t ∈T. We 9nd the followingresult:

Theorem 3.5. If p is rd-continuous and real; then

sinGp = cosp; cosG

p = − sinp; and sin2p + cos2

p = e�p2 :

In particular, if a∈R is constant, then cosa(·; t0) solves the initial value problem

yG2= − ay; y(t0) = 1; yG(t0) = 0:

Finally, we mention the analogue of Theorem 3.1 in the matrix case. This result can also be used tostudy higher order equations by rewriting them 9rst in matrix form. Let P be an n×n-matrix-valued,rd-continuous (i.e., each entry is rd-continuous) function of T. We say that P is regressive provided

I + �(t)P(t) is invertible for all t ∈T;


Table 1The two most important examples

Time scale T R Z

Backward jump operator (t) t t − 1Forward jump operator �(t) t t + 1Graininess �(t) 0 1Derivative fG(t) f′(t) Gf(t)

Integral∫ ba f(t)Gt

∫ ba f(t) dt

∑b−1t=a f(t) (if a¡b)

Rd-continuous f continuous f any f�-operator I� I I\{maxI}

where I denotes the n× n-identity matrix. Concerning initial value problems of the form

YG =P(t)Y; Y (t0) = I (3.5)

we have the following result of Hilger [12]:

Theorem 3.6. If P is rd-continuous and regressive; then (3:5) has a unique solution.

We refer to this solution as the matrix exponential and denote it by eP(·; t0). In the case of aconstant matrix A it is possible to calculate eA(·; t0) using a generalization of Putzer’s algorithm asis explained in [3].

4. Examples and applications

Example 4.1. Of course the two most popular examples of time scales are R and Z. In Table 1 wecollect some information about these time scales.

Now let us solve (3.1) for these two cases and hence 9nd the corresponding exponential functions:Clearly, if T=R, then

ep(t; s) = exp{∫ t

sp(�) d�

}and ea(t; s) = ea(t−s)

if p is continuous and a is constant. Hence e.g.,

ea(t; 0) = eat and e1(t; 0) = et:

Similarly,

sinhp(t; s) = sinh{∫ t

sp(�) d�

}; sinha(t; s) = sinh (a(t − s));

and hence e.g.,

sinha(t; 0) = sinh (at) and sinh1(t; 0) = sinh t;


and corresponding formulas can be written down for cosh, sin, and cos. The functions hk and gk aregiven by

hk(t; s) = gk(t; s) =(t − s)k

k!; e:g:; hk(t; 0) = gk(t; 0) =

tk

k!:

Next, if T=Z, then (3.1) reads

Gy(t) :=y(t + 1) − y(t) =p(t)y(t + 1); y(t0) = 1:

Hence, if p(t) = − 1 for all t ∈Z, then

ep(t; s) =

∏t−1

�=s[1 + p(�)] if t ¿ s

1 if t= s

1∏s−1�=t [1 + p(�)]

if t ¡ s

and therefore with constant a = − 1

ea(t; s) = (1 + a)t−s; ea(t; 0) = (1 + a)t ; e1(t; 0) = 2t :

Moreover, e.g.,

cosha(t; 0) =(1 + a)t + (1 − a)t

2=

∞∑k=0

(t

2k + 1

)a2k+1

and

sinha(t; 0) =(1 + a)t − (1 − a)t

2=

∞∑k=0

(t

2k

)a2k

and similarly for the trigonometric functions. The functions hk and gk are given by

hk(t; s) =(t − sk

)and gk(t; s) =

(t − s + k − 1

k

)if t ¿ s:

Example 4.2. Another important time scale is

hZ= {hk | k ∈Z} for some h¿ 0:

Dynamic equations on hZ correspond to di6erence equations with step size h rather than 1. Hence(3.1) reads

y(t + h) − y(t)h

=p(t)y(t); y(t0) = 1:

E.g., if h= 1=n for some natural number n∈N; then

ea(t; 0) =(

1 +an

)nt → eat as n → ∞:


Many properties on this time scale “become” the corresponding “continuous” properties as h → 0.As one more example we use Theorem 3.5 to 9nd

sin2a(t; 0) + cos2

a(t; 0) = ea2=n(t; 0) =(

1 +a2

n2

)nt→ 1 as n → ∞:

The use of hZ lies in the “discretization” aspect of the time scales calculus. Time scales which servethis purpose even better are of the form

T= {tk(h) | k ∈Z} with 06 tk+1(h) − tk(h) = o(h) as h → 0:

They allow discretization with variable step size.

Example 4.3. Let N (t) be the amount of plants of one particular kind at time t in a certain area. Byexperiments we know that N grows exponentially according to N ′ =N during the months of Apriluntil September. At the beginning of October, all plants suddenly die, but the seeds remain in theground and start growing again at the beginning of April with N now being doubled. We modelthis situation using the time scale

T=∞⋃k=0

[2k; 2k + 1];

where t= 0 is April 1 of the current year, t= 1 is October 1 of the current year, t= 2 is April 1 ofthe next year, t= 3 is October 1 of the next year, and so on. We have

�(t) =

{0 if 2k6 t ¡ 2k + 1

1 if t= 2k + 1:

On [2k; 2k+1); we have N ′ =N; i.e., NG =N . However, we also know that N (2k+2) = 2N (2k+1);i.e., GN (2k + 1) =N (2k + 1); i.e., NG =N at 2k + 1. As a result, N is a solution of the dynamicequation

NG =N:

Thus, if N (0) = 1 is given, N is exactly e1(·; 0) on the time scale T. Further examples that canbe modeled with similar time scales include insect population models, which are discrete in season(and may follow a di6erence scheme with variable step size or are often modeled by continuousdynamic systems), die out in say winter, while their eggs are incubating or dormant, and then inseason again, hatching gives rise to a nonoverlapping population. Some speci9c examples [9] arethe cicada Magicicada septendecim which lives as a larva for 17 years and as an adult for perhapsa week, and the common mayQy Stenonema canadense which lives as a larva for a year and as anadult for less than a day.

Example 4.4 (S. Keller [15]). Consider a simple electric circuit with resistance R; inductance L;and capacitance C. Suppose we decharge the capacitor periodically every time unit and assume thatthe decharging takes -¿ 0 (but small) time units. Then this stimulation can be modeled using thetime scale T=

⋃k∈N0

[k; k + 1− -]. If Q(t) is the total charge on the capacitor at time t and I(t) is


the current as a function of time t; then the corresponding dynamic system that is satis9ed by Qand I (on T) reads as follows:

QG =

bQ on

⋃k∈N

{k − -}

I otherwise;and IG =

0 on⋃k∈N

{k − -}

− 1LCQ − R

L I otherwise:

Example 4.5. So-called q-di8erence equations, q¿ 1 have been studied extensively in the literature,see e.g., [4,16,17]. An example of such an equation is

y(qt) − y(t) = (q− 1)t:

The time scale, which has such q-di6erence equations as its dynamic equations, is given by

qN0 = {qk | k ∈N0} for some q¿ 1;

where N0 =N ∪ {0}. We have �(t) = qt for all t ∈ qN0 . If t0 = 1; then (3.1) reads

y(qt) − y(t)(q− 1)t

=p(t)y(t); y(1) = 1:

This yields

ep(qk ; 1) =k−1∏/=0

[1 + (q− 1)q/p(q/)] for all k ∈N:

In particular, if p(s) = (1 − s)=(q− 1)s2; then for t= qk

e1(t; 1) = e1(qk ; 1) =k−1∏/=0

[1 +

1 − q/

q/

]=

k−1∏/=0

1q/

=1

qk(k−1)=2

= q−k2=2qk=2 =

√qk exp

{−k

2 log q2

}=√qk exp

{−(k log q)2

2 log q

}

=√t exp

{− log2 t

2 log q

}:

Moreover, in [2] we 9nd

hk(t; s) =k−1∏/=0

t − q/s∑/�=0 q

� if t ¿ s:

E.g., if q= 2; we have

h2(t; 1) =(t − 1)(t − 2)

3; h3(t; 1) =

(t − 1)(t − 2)(t − 4)21

;

and

h4(t; 1) =(t − 1)(t − 2)(t − 4)(t − 8)

315:


Example 4.6. More generally, let {�k}k∈N be a sequence of positive numbers and let t0 ∈R. Let

tk = t0 +k∑/=1

�/ for k ∈N:

Put

T=

{ {tk | k ∈N0} if limk→∞ tk =∞{tk | k ∈N0} ∪ {t∗} if limk→∞ tk = t∗:

Then

�(tk) = tk+1 and �(tk) = �k+1 for all k ∈N0:

Problem (3.1) now reads

y(tk+1) − y(tk)�k+1

=p(tk)y(tk); y(t0) = 1;

and hence has the solution

ep(tk ; t0) =k∏/=1

[1 + �/p(t/−1)] for all k ∈N0: (4.1)

Now we consider some more speci9c examples:(i) Let t0 = 1 and �k = (1 − 1=q)qk for k ∈N. Then

tk = t0 +k∑/=1

�/ = 1 +(

1 − 1q

) k∑/=1

q/ = 1 +q− 1q

k−1∑/=0

q/+1

= 1 + (q− 1)k−1∑/=0

q/ = 1 + (q− 1)qk − 1q− 1

= qk ;

so T= qN0 . By (4.1), the solution of (3.1) is given by

ep(qk ; 1) =k∏/=1

[1 +(

1 − 1q

)q/p(q/−1)

]=

k−1∏/=0

[1 + (q− 1)q/p(q/)];

compare with Example 4.5.(ii) Let t0 = 0 and �k = 1=k for k ∈N. Then

T=

{k∑/=1

1/

∣∣∣∣∣ k ∈N}

∪ 0:

Suppose p(t) ≡ N ∈N is constant. By (4.1), the solution of (3.1) is given by

eN (tk ; 0) =k∏/=1

(1 +

N/

)=

k∏/=1

N + //

=(N + k)!N !k!

=(N + kk

):


(iii) Let t0 = 1 and �k = 2k + 1 for k ∈N. Then

tk = t0 +k∑/=1

�/ = 1 +k∑/=1

(2/+ 1) = 1 + 2k(k + 1)

2+ k = (k + 1)2

and

T=N2: = {k2 | k ∈N}:Suppose p(t) ≡ 1. By (4.1), the solution of (3.1) is given by

e1((k + 1)2; 1) = e1(tk ; 1) =k∏/=1

(1 + �/) =k∏/=1

(2 + 2/) = 2kk∏/=1

(1 + /) = 2k(k + 1)!

so that

e1(t; 1) = 2√t−1(

√t)! for all t ∈T:

(iv) Let t0 = 0 and �k = 1=(4k2 − 1) for k ∈N. In this case

tk =k∑/=1

�/ =k∑/=1

14/2 − 1

converges, say to t∗. Suppose p(t) ≡ 1. By (4.1),

e1(tk ; 0) =k∏/=1

(1 + �/) =k∏/=1

(1 +

14/2 − 1

)=

k∏/=1

4/2

4/2 − 1

solves (3:1) and

limt→t∗

e1(t; 0) = limk→∞

{k∏/=1

4/2

4/2 − 1

}=02

by a well-known identity on this so-called Wallis product.

5. Linear dynamic equations

Clearly, using Theorem 3.3, the unique solution of

xG = − p(t)x�; x(t0) = x0

is x0e�p(·; t0); provided p is regressive and rd-continuous. If x solves

xG = − p(t)x� + f(t); x(t0) = x0 (5.1)


where, in addition, f is rd-continuous, then we multiply the dynamic equation in (5.1) by theintegrating factor ep(t; t0) to obtain

[xep(·; t0)]G = xGep(·; t0) + pep(·; t0)x� = ep(·; t0)[xG + px�] = ep(·; t0)f:We integrate both sides between t0 and t to arrive at a formula for x. Using (2.2), we also can solvethe problem

yG =p(t)y + f(t); y(t0) =y0: (5.2)

Hence we obtain the following result from [8]:

Theorem 5.1. If p and f are rd-continuous and p is regressive, then

x0e�p(t; t0) +∫ t

t0

e�p(t; �)f(�)G� and y0ep(t; t0) +∫ t

t0

ep(t; �(�))f(�)G�

solve (5:1) and (5:2); respectively.

Now we consider second order equations with constant coe:cients

yGG + �yG + �y= 0: (5.3)

Trying y= e�(·; t0) with some regressive constant �; we 9nd that y solves (5.3) provided � satis9esthe characteristic equation

�2 + �� + �= 0;

which has solutions

�1 =−�−

√�2 − 4�

2and �2 =

−� +√�2 − 4�

2:

The Wronskian of y1 = e�1(·; t0) and y2 = e�2(·; t0) is de9ned as

W (y1; y2) = det

(y1 y2

yG1 yG

2

)

and calculates to

W (y1; y2) = e�1(·; t0)�2 e�2(·; t0) − �1 e�1(·; t0) e�2(·; t0)= (�2 − �1) e�1⊕�2(·; t0):

Hence W is never zero if �1 = �2; i.e., if �2 = 4�. In this case we say that y1 and y2 form afundamental system of (5.3), and the solution of any initial value problem

yGG + �yG + �y= 0; y(t0) =y0; yG(t0) =yG0 (5.4)


can be given as a linear combination of y1 and y2. Now we can state the main result on equationsof the form (5.3), see [8].

Theorem 5.2. (i) If �2−4�¿ 0; put p=−�=2 and q=√�2 − 4�=2. If p and ��−� are regressive;

then a fundamental system of (5.3) is given by coshq=(1+�p)(·; t0) ep(·; t0); sinhq=(1+�p)(·; t0)ep(·; t0);with Wronskian qe��−�(·; t0); and the solution of (5.4) is[

y0 coshq=(1+�p)(·; t0) +yG

0 − py0

qsinhq=(1+�p)(·; t0)

]ep(·; t0):

(ii) If �2 − 4�¡ 0; put p= − �=2 and q=√

4� − �2=2. If p; q∈R and p is regressive; thena fundamental system of (5.3) is given by cosq=(1+�p)(·; t0) ep(·; t0); sinq=(1+�p)(·; t0)ep(·; t0); withWronskian qe��−�(·; t0); and the solution of (5.4) is[

y0 cosq=(1+�p)(·; t0) +yG

0 − py0

qsinq=(1+�p)(·; t0)

]ep(·; t0):

(iii) If �2 −4�= 0; put p=−�=2. If p is regressive; then a fundamental system of (5.3) is givenby ep(t; t0) and ep(t; t0)

∫ tt0

G�1+p�(�) ; with Wronskian e�p2(·; t0); and the solution of (5.4) is

[y0 + (yG

0 − py0)∫ t

t0

G�1 + p�(�)

]ep(t; t0):

Another result concerns the Euler–Cauchy equation

t�(t)yGG + �tyG + �y= 0; t ¿ 0; (5.5)

where � and � are constant, and where we assume that the regressivity condition

1 − ��(t)�(t)

+��2(t)t�(t)

= 0 (5.6)

holds. The associated characteristic equation is

�2 + (�− 1)� + �= 0: (5.7)

Theorem 5.3. Suppose (5.6). If (5.7) has two distinct roots �1 and �2; then

e�1=t(t; t0) and e�2=t(t; t0)

is a fundamental system of (5.5). If (�− 1)2 = 4b2; then we put p= (�− 1)=2; and

ep=t(t; t0) and ep=t(t; t0)∫ t

t0

G��+ p�(�)

is a fundamental system of (5.5).


For the more general problem

yGG + p(t)yG + q(t)y=f(t); y(t0) =y0; yG(t0) =yG0 (5.8)

we have the following variation of constants result:

Theorem 5.4. If f is rd-continuous and if y1 and y2 form a fundamental system for the equationyGG + p(t)yG + q(t)y= 0; then the solution of (5.8) is given by

c1y1(t) + c2y2(t) +∫ t

t0

y2(t)y1(�(�)) − y1(t)y2(�(�))W (y1; y2)(�(�))

f(�)G�;

where

c1 =yG

2 (t0)y0 − y2(t0)yG0

W (y1; y2)(t0)and c2 =

y1(t0)yG0 − yG

1 (t0)y0

W (y1; y2)(t0):

Let us now consider the Wronskian of any two solutions x1 and x2 of

xGG + p(t)xG�+ q(t)x� = 0: (5.9)

We have

W (x1; x2)G = det

(x1 x2

xG1 xG

2

)G

= (x1xG2 − xG

1 x2)G

= x�1xGG2 + xG

1 xG2 − xGG

1 x�2 − xG1 x

G2

= x�1xGG2 − x�2x

GG1 = det

(x�1 x�2

xGG1 xGG

2

)

= det

(x�1 x�2

−pxG�

1 −qx�1 −pxG�

2 −qx�2

)

= det

(x�1 x�2

−pxG�

1 −pxG�

2

)= − p det

(x�1 x�2

xG�

1 xG�

2

)

=−p det

(x1 x2

xGG1 xGG

2

)= − pW (x1; x2)�:

Hence, by Theorem 5.1, we obtain the following Abel’s formula:

Theorem 5.5. If p is regressive and rd-continuous and if x1 and x2 solve (5.9), then

W (x1; x2)(t) =W (x1; x2)(t0)e�p(t; t0):

A similar result holds for higher order equations of the form

xGn+

n∑k=1

qk(t)(xGn−k

)�= 0: (5.10)


It can be shown that initial value problems involving equation (5.10) have unique solutions providedthe qk are rd-continuous for 16 k6 n and q1 is regressive, see [6]. Wronskians of n functionsx1; : : : ; xn; all of them n− 1 times di6erentiable, are de9ned in the natural way

W (x1; : : : ; xn) = det

x1 · · · xn

xG1 · · · xG

n

......

xGn−1

1 · · · xGn−1

n

: (5.11)

Theorem 5.6. If q1 is regressive and rd-continuous and if x1; : : : ; xn solve (5.10), then

W (x1; : : : ; xn)(t) =W (x1; : : : ; xn)(t0)e�q1(t; t0):

The variations of constant result for higher order equations is stated in terms of

yGn+

n∑k=1

pk(t)yGn−k=f(t) (5.12)

and reads as follows:

Theorem 5.7. If f;p1; : : : ; pn are rd-continuous and −∑nk=1(−�)k−1pk is regressive; and if y1; : : : ; yn

is a fundamental system for yGn+∑n

k=1 pk(t)yGn−k

= 0; then all solutions of (5.12) are given byn∑

k=1

ckyk(t) +∫ t

t0

W (�(�); t)W (�(�))

f(�)G�;

where c1; : : : ; cn are constants; W =W (y1; : : : ; yn); and W (�; t) is the determinant of the matrix in(5.11) where the last row is replaced by (y1(t) · · ·yn(t)).

For the remainder of this section we now consider self-adjoint equations of second order

(p(t)xG)G + q(t)x� = 0 with p(t) = 0 for all t ∈T: (5.13)

Clearly, by Theorem 5.5, if p(t) ≡ 1; then the Wronskian of any two solutions of (5.13) is constant.In case of (5.13), this is true for the generalized Wronskian

W (x1; x2) = det

(x1 x2

pxG1 pxG

2

)

since

W (x1; x2)G = (x1pxG2 − x2pxG

1 )G = x�1 (pxG2 )G − x�2 (pxG

1 )G = − x�1qx�2 + x�2qx

�1 = 0:

Theorem 5.8. The Wronskian of any two solutions of (5.13) is constant.


Now assume that x solves (5.13) such that x(t) = 0 for all t ∈T. We make the Riccati substitution

r=pxG

x

and 9nd that

rG + q +r2

p + �r=

(pxG)Gx − xGpxG

xx�+ q +

p2(xG)2

x2

/(p +

p�xG

x

)

=−qx�x − p(xG)2

xx�+ q +

p2(xG)2

x2

xpx�

= 0:

Hence r solves the Riccati equation

rG + q(t) +r2

p(t) + �(t)r= 0: (5.14)

The following Riccati equivalence is shown in [1]:

Theorem 5.9. There exists a solution x of (5.13) with x(t) = 0 for all t ∈T i8 (5.14) has a solutionr (related by r=pxG=x).

Now suppose again that x solves (5.13) and is never zero. Let 3 be any di6erentiable function onT. With r=pxG=x we have

(32r)G + (p3G − r3)2 xpx�

=(32

x

)�(pxG)G +

(32

x

)G

pxG + p(3G − xG3

x

)2 xx�

= −(32

x

)�qx� +

[3G 3

x+ 3�

(3x

)G]pxG + p

[(3x

)G]2

xx�

= − (3�)2q +33GpxG

x+ p

(3x

)G [3�xG +

(3x

)Gxx�]

= − (3�)2q +33GpxG

x+ p

(3x

)G[3�xG + 3Gx − 3xG]

= − (3�)2q +33GpxG

x+ p

(3x

)G3Gx�

=p(3G)2 − q(3�)2:

This so-called Picone identity can be used to prove the following Jacobi condition which is a resulton positive de@niteness of the quadratic functional

F(3) =∫ b

a{p(3G)2 − q(3�)2}(t)Gt where a; b∈T:


Here, F is called positive de9nite if F(3)¿ 0 for all nontrivial di6erentiable functions 3 with3(a) = 3(b) = 0. We also call (5.13) disconjugate (on [a; b]) if the solution x of

(p(t)xG)G + q(t)x� = 0; x(a) = 0; xG(a) =1

p(a)

satis9es

px�x¿ 0 on (a; b]�:

Theorem 5.10. F is positive de@nite i8 (5.13) is disconjugate.

Finally, we present a Sturm separation theorem and a Sturm comparison theorem. Sturmian theorycan be developed as follows: We say that a solution x of (5.13) has no focal points in (a; b] if

px�x¿ 0 on (a; b]� and p(a)x(a)x�(a)¿ 0:

Theorem 5.11. If there exists a solution of (5.13) without focal points; then (5:13) is disconjugate.

Consider another self-adjoint equation of the form (5.13),

(p(t)xG)G + q(t)x� = 0: (5.15)

Theorem 5.12. Suppose p(t)6p(t) and q(t)¿ q(t) for all t ∈T. If (5.14) is disconjugate; thenso is (5:15).

6. Symplectic systems

Let us now rewrite (5.13) as a system by introducing u=pxG. We 9nd

xG =1pu and uG = − qx� = − qx − �q

pu;

so with z= ( xu) we have

zG = S(t)z where S =

(0 1

p

−q −�qp

):

The matrix I + �S has determinant

det

1 �

p

−�q 1 − �2qp

= 1 − �2q

p+�2qp

= 1

and hence S is regressive. Subject to the assumptions that p and q are rd-continuous, Theorem3.6 yields the unique solvability of initial value problems involving zG = S(t)z. The above


2 × 2-matrix-valued function S has, besides being rd-continuous and regressive, another remarkable

property, namely, we have with T=

(0 1

−1 0

)

STT + TS + �STTS =

(q 0�qp

1p

)+

(−q −�qp

0 − 1p

)+ �

(q 0�qp

1p

)(0 1

p

−q −�qp

)

=

(0 −�q

p

�qp 0

)+ �

(0 q

p

− qp 0

)=

(0 0

0 0

);

i.e.,

STT + TS + �STTS = 0: (6.1)

In general, we call a 2n× 2n-matrix-valued function S symplectic (with respect to T) if (6.1) holdswith

T=

(0 I

−I 0

):

The corresponding system

zG = S(t)z where S satis9es (6:1) and is rd-continuous (6.2)

is called a symplectic system. If S is symplectic (w.r.t. T), then it is regressive:

(I + �S)TT(I + �S) =T + �(STT + TS + �STTS) =T: (6.3)

Hence, by Theorem 3.6, initial value problems involving symplectic systems (6.2) have uniquesolutions. Now let z1 and z2 be any two solutions of (6.2). Then their (generalized) Wronskian isde9ned as

W (z1; z2) = zT1 Tz2:

Because of

W (z1; z2)G = (zG1 )TTz�2 + zT1 TzG

2

= (zG1 )TT(z2 + �zG

2 ) + zT1 TzG2

= zT1 STT(z2 + �Sz2) + zT1 TSz2

= zT1 [STT + TS + �STTS]z2

= 0;

we have, see [10]:

Theorem 6.1. The Wronskian of any two solutions of (6:2) is constant.

A conjoined solution of (6.2) is a 2n× n-matrix-valued solution Z of (6.2) with ZTTZ ≡ 0. Theconjoined solution of (6.2) satisfying Z(a) = (0

I ) is called the principal solution of (6.2) (at a).


Example 6.1. First, S is symplectic with respect of R i6

STT + TS = 0;

i.e., i6 TS is symmetric. Then, necessarily, S must be of the form

S =

(A B

C −AT

)with symmetric B; C

and is called Hamiltonian. Next, S is symplectic with respect to Z i6, see (6.3),

(I + S)TT(I + S) =T;

i.e., i6 I + S is symplectic. Here, as usual, a matrix 2n × 2n-matrix M is called symplectic, ifMTTM =T. Note also that (6.3) shows that I + �S is symplectic whenever S is symplectic withrespect to T. We also remark that symplectic systems and Hamiltonian systems are the same forT=R but that there are more discrete symplectic systems than Hamiltonian di6erence systems,see [5].

Now, let us assume that (6.2) has a conjoined solution Z = ( XU ) such that X is invertible. Weconsider the Riccati substitution

R=UX−1:

The calculation

RG =UG(X �)−1 − UX−1XG(X �)−1

= (UG − RXG)(X �)−1

= (I R)TZG(X �)−1

= (I R)TSZ(X �)−1

= (I R)TS(I + �S)−1Z�(X �)−1

=−(I R)STT(IR�

)

shows that R is a (symmetric) solution of the matrix Riccati equation

RG + (I R)STT(IR�

)= 0:

The precise statement of the Riccati equivalence (compare Theorem 5.9) may be found in [47,Theorem 3]. In [14], the generalization of Theorem 5.10 is derived:

Theorem 6.2. Subject to a normality condition, F is positive de@nite i8 (6.2) is disconjugate.


In Theorem 6.2, the quadratic functional

F(z) =∫ b

a{zT (STMS −M)z}(t)Gt with M=

(0 0

I 0

)

is called positive de9nite, if F(z)¿ 0 for all z with MzG =MSz; Mz ≡ 0, and Mz(a) =Mz(b) = 0.System (6.2) is called disconjugate if the principal solution of (6.2) (at a) has no focal points in(a; b], and this is de9ned as{

X (t) is invertible if t ∈ (a; b] is left-dense or right-denseKer X (�(t)) ⊂ Ker X (t); X (t)X †(�(t))B(t)¿ 0 for all t ∈ (a; b]�:

Here, Ker stands for the kernel and the dagger denotes the Moore–Penrose inverse of the matrixindicated, “¿ 0” means positive semide9nite, and B is the n × n-matrix in the right upper cornerof S.

References

[1] R.P. Agarwal, M. Bohner, Quadratic functionals for second order matrix equations on time scales, Nonlinear Anal.33 (1998) 675–692.

[2] R.P. Agarwal, M. Bohner, Basic calculus on time scales and some of its applications, Results Math. 35 (1–2) (1999)3–22.

[3] C.D. Ahlbrandt, J. Ridenhour. Putzer algorithms for matrix exponentials, matrix powers, and matrix logarithms, 2000,in preparation.

[4] J.P. BUezivin, Sur les Uequations fonctionnelles aux q-di6Uerences, Aequationes Math. 43 (1993) 159–176.[5] M. Bohner, O. DoVslUy, Disconjugacy and transformations for symplectic systems, Rocky Mountain J. Math. 27 (3)

(1997) 707–743.[6] M. Bohner, P.W. Eloe, Higher order dynamic equations on measure chains: Wronskians, disconjugacy, and

interpolating families of functions, J. Math. Anal. Appl. 246 (2000) 639–656.[7] M. Bohner, A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, BirkhWauser, Boston,

2001.[8] M. Bohner, A. Peterson, First and second order linear dynamic equations on measure chains, J. Di6er. Equations

Appl. 2001, to appear.[9] F.B. Christiansen, T.M. Fenchel, Theories of Populations in Biological Communities, Lecture Notes in Ecological

Studies, vol. 20, Springer, Berlin, 1977.[10] O. DoVslUy, R. Hilscher, Disconjugacy, transformations and quadratic functionals for symplectic dynamic systems on

time scales, J. Di6er. Equations Appl. 2001, to appear.[11] S. Hilger, Ein MaXkettenkalkWul mit Anwendung auf Zentrumsmannigfaltigkeiten. Ph.D. thesis, UniversitWat WWurzburg,

1988.[12] S. Hilger, Analysis on measure chains—a uni9ed approach to continuous and discrete calculus, Results Math. 18

(1990) 18–56.[13] S. Hilger, Special functions, Laplace and Fourier transform on measure chains, Dynam. Systems Appl. 8(3–4)

(1999) 471–488 (Special Issue on “Discrete and Continuous Hamiltonian Systems”, edited by R.P. Agarwal andM. Bohner).

[14] R. Hilscher, Reid roundabout theorem for symplectic dynamic systems on time scales, Appl. Math. Optim. 2001, toappear.

[15] S. Keller, Asymptotisches Verhalten invarianter FaserbWundel bei Diskretisierung und Mittelwertbildung im Rahmender Analysis auf Zeitskalen. Ph.D. thesis, UniversitWat Augsburg, 1999.

[16] W.J. Trijtzinsky, Analytic theory of linear q-di6erence equations, Acta Math. 61 (1933) 1–38.[17] C. Zhang, Sur la sommabilitUe des sUeries enti[eres solutions d’Uequations aux q-di6Uerences, I., C.R. Acad. Sci. Paris

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Further reading

[1] R. Agarwal, C. Ahlbrandt, M. Bohner, A. Peterson, Discrete linear Hamiltonian systems: A survey. Dynam. Sys-tems Appl. 8(3–4) (1999) 307–333 (Special Issue on “Discrete and Continuous Hamiltonian Systems”, edited byR.P. Agarwal and M. Bohner).

[2] R. Agarwal, M. Bohner, A. Peterson, Inequalities on time scales: A survey. 2001. Submitted for publication.[3] R.P. Agarwal, Di6erence Equations and Inequalities, Marcel Dekker, Inc., New York, 1992.[4] R.P. Agarwal, M. Bohner, D. O’Regan, Time scale boundary value problems on in9nite intervals, this issue, J.

Comput. Appl. Math. 141 (2002) 27–34.[5] R.P. Agarwal, M. Bohner, D. O’Regan, Time scale systems on in9nite intervals, Nonlinear Anal. 2001, to appear.[6] R.P. Agarwal, M. Bohner, P.J.Y. Wong, Sturm–Liouville eigenvalue problems on time scales, Appl. Math. Comput.

99 (1999) 153–166.[7] R.P. Agarwal, D. O’Regan, Triple solutions to boundary value problems on time scales, Appl. Math. Lett. 13 (4)

(2000) 7–11.[8] R.P. Agarwal, D. O’Regan, Nonlinear boundary value problems on time scales, Nonlinear Anal. 2001, to appear.[9] C.D. Ahlbrandt, M. Bohner, J. Ridenhour, Hamiltonian systems on time scales, J. Math. Anal. Appl. 250 (2000)

561–578.[10] C.D. Ahlbrandt, A.C. Peterson, Discrete Hamiltonian Systems: Di6erence Equations, Continued Fractions, and Riccati

Equations, Kluwer Texts in the Mathematical Sciences, vol. 16, Kluwer Academic Publishers, Boston, 1996.[11] E. Ak]n, Boundary value problems for a di6erential equation on a measure chain, Panamer. Math. J. 10 (3) (2000)

17–30.[12] D. Anderson, A. Peterson, Asymptotic properties of solutions of a 2nth-order di6erential equation on a time scale,

Math. Comput. Modelling 32(5–6) (2000) 653–660 (Boundary value problems and related topics).[13] F.M. At]c], G.Sh. Guseinov, B. Kaymakcalan, On Lyapunov inequality in stability theory for Hill’s equation on time

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